Hello,
First off, if you’re using Appearance > Background, you don’t have to have your background image tiled. WordPress has a setting there you can choose on how to repeat the background image. You can choose to have it not repeat and be centered.
And if you were going to use a plugin that doesn’t properly add a background into the WordPress system, you’d have to hide the theme’s background on the #wrapper div manually with CSS Edits.
For CSS changes, it’s a good idea to use a tool like firebug or Google Chrome’s built-in developer tools to locate them and then put the edits either in your child theme or in the “Custom CSS” option on your theme options page.
#wrapper { background-image:none; }
Also, here’s a helpful tutorial on how to use Firebug – http://vimeo.com/20917974. The only thing I’d suggest not doing is actually making edits to the theme’s CSS files, as it shows at the end of that video. However, the video serves as a good introduction to Firebug and how it can be used to find what CSS is effecting certain areas of your site without ever digging around in any files.